In this section we briefly demonstrate the numerical scheme of time-domain integration, first proposed in [1]}, which yields the temporal response of the scalar field as it propagates on a fixed background. By defining \(\psi (\rho _\ast ,t)=\psi (i\Delta \rho _\ast ,j\Delta t)=\psi _{i,j}\) , \(V(\rho (\rho _*))=V(\rho _\ast ,t)=V(i\Delta \rho _\ast ,j\Delta t)=V_{i,j}\) , equation (REF ) takes the form
\(\frac{\psi _{i+1,j} - 2\psi _{i,j} + \psi _{i-1,j} }{\Delta \rho ^2_\ast } - \frac{ \psi _{i,j+1} - 2\psi _{i,j} + \psi _{i,j-1} }{\Delta t^2} - V_i \psi _{i,j} = 0\,.\) 
